## Some asymptotic laws for crossings and excursions.(English)Zbl 0666.60070

Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 59-74 (1988).
[For the entire collection see Zbl 0649.00017.]
Working with a recurrent Hunt process having an invariant reference measure, the authors consider various processes associated with a pair $$A, C$$ of closed subsets of the state space such that $$A\cap C$$ is polar, or with $$M$$, a closed optimal subset of $$(0,\infty)$$. For the counting processes for the number of “crossings” from $$A$$ to $$C$$ in $$(0,t]$$, or for the number of excursions outside of $$M$$ of some fixed type in $$(0,t]$$, general asymptotic results are deduced from an ergodic theorem for additive processes.
The Markov chains of starting points of crossings or of excursions of a certain type are described. Asymptotics for crossings of planar Brownian motion, both in case the logarithmic capacity of $$A$$ relative to $$C$$ is finite (e.g., $$A$$ and $$C$$ are non-intersecting circles) and in case it is infinite (as when $$A\cap C$$ consists of a finite number of points) are derived. The analysis for excursions is related to the notion of Palm measure associated with a stationary process of excursions.

### MSC:

 60J25 Continuous-time Markov processes on general state spaces 60J55 Local time and additive functionals 60J65 Brownian motion

Zbl 0649.00017